Theorem difopab 5854

Description: Difference of two ordered-pair class abstractions. (Contributed by Stefan O'Rear, 17-Jan-2015.) (Proof shortened by SN, 19-Dec-2024.)

Assertion

Ref	Expression
difopab	⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ ¬ 𝜓)}

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem difopab

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relopabv 5845	. . 3 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		reldif 5839	. . 3 ⊢ (Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑} → Rel ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓}))
3	1, 2	ax-mp 5	. 2 ⊢ Rel ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓})
4		relopabv 5845	. 2 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ ¬ 𝜓)}
5		sban 2080	. . . 4 ⊢ ([𝑧 / 𝑥]([𝑤 / 𝑦]𝜑 ∧ [𝑤 / 𝑦] ¬ 𝜓) ↔ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦] ¬ 𝜓))
6		sban 2080	. . . . 5 ⊢ ([𝑤 / 𝑦](𝜑 ∧ ¬ 𝜓) ↔ ([𝑤 / 𝑦]𝜑 ∧ [𝑤 / 𝑦] ¬ 𝜓))
7	6	sbbii 2076	. . . 4 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦](𝜑 ∧ ¬ 𝜓) ↔ [𝑧 / 𝑥]([𝑤 / 𝑦]𝜑 ∧ [𝑤 / 𝑦] ¬ 𝜓))
8		vopelopabsb 5548	. . . . 5 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
9		sbn 2284	. . . . . 6 ⊢ ([𝑧 / 𝑥] ¬ [𝑤 / 𝑦]𝜓 ↔ ¬ [𝑧 / 𝑥][𝑤 / 𝑦]𝜓)
10		sbn 2284	. . . . . . 7 ⊢ ([𝑤 / 𝑦] ¬ 𝜓 ↔ ¬ [𝑤 / 𝑦]𝜓)
11	10	sbbii 2076	. . . . . 6 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦] ¬ 𝜓 ↔ [𝑧 / 𝑥] ¬ [𝑤 / 𝑦]𝜓)
12		vopelopabsb 5548	. . . . . . 7 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜓)
13	12	notbii 320	. . . . . 6 ⊢ (¬ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ¬ [𝑧 / 𝑥][𝑤 / 𝑦]𝜓)
14	9, 11, 13	3bitr4ri 304	. . . . 5 ⊢ (¬ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ [𝑧 / 𝑥][𝑤 / 𝑦] ¬ 𝜓)
15	8, 14	anbi12i 627	. . . 4 ⊢ ((⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ ¬ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦] ¬ 𝜓))
16	5, 7, 15	3bitr4ri 304	. . 3 ⊢ ((⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ ¬ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ [𝑧 / 𝑥][𝑤 / 𝑦](𝜑 ∧ ¬ 𝜓))
17		eldif 3986	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ ¬ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}))
18		vopelopabsb 5548	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ ¬ 𝜓)} ↔ [𝑧 / 𝑥][𝑤 / 𝑦](𝜑 ∧ ¬ 𝜓))
19	16, 17, 18	3bitr4i 303	. 2 ⊢ (⟨𝑧, 𝑤⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ ¬ 𝜓)})
20	3, 4, 19	eqrelriiv 5814	1 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ ¬ 𝜓)}

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1537  [wsb 2064   ∈ wcel 2108   ∖ cdif 3973  ⟨cop 4654  {copab 5228  Rel wrel 5705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-opab 5229  df-xp 5706  df-rel 5707

This theorem is referenced by:  dfnelbr2  47188